Space of modular forms of level 242, weight 2, and trivial character

• Label: 242.2.a
• Level: $242$
• Weight: $2$
• Character orbit: 242.a
• Rep. character: $\chi_{242}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $6$
• Sturm bound: $66$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	242.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(66\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(242))\).

	Total	New	Old
Modular forms	45	10	35
Cusp forms	22	10	12
Eisenstein series	23	0	23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(3\)
\(-\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	$+$	\(1\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(7\)

Trace form

\( 10 q - 2 q^{3} + 10 q^{4} - 2 q^{5} + 8 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(242))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	11
242.2.a.a	$242$	$2$	242.a	1.a	$1$	$1$	$1$	$1.932$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-2\)	\(-3\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}-3q^{5}+2q^{6}+2q^{7}+\cdots\)
242.2.a.b	$242$	$2$	242.a	1.a	$1$	$1$	$1$	$1.932$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-2\)	\(-3\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-3q^{5}-2q^{6}-2q^{7}+\cdots\)
242.2.a.c	$242$	$2$	242.a	1.a	$1$	$2$	$2$	$1.932$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta )q^{3}+q^{4}-\beta q^{5}+(1+\cdots)q^{6}+\cdots\)
242.2.a.d	$242$	$2$	242.a	1.a	$1$	$2$	$2$	$1.932$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-2\)	\(3\)	\(2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta )q^{3}+q^{4}+(2-2\beta )q^{5}+\cdots\)
242.2.a.e	$242$	$2$	242.a	1.a	$1$	$2$	$2$	$1.932$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta )q^{3}+q^{4}-\beta q^{5}+(-1+\cdots)q^{6}+\cdots\)
242.2.a.f	$242$	$2$	242.a	1.a	$1$	$2$	$2$	$1.932$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(3\)	\(2\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+(2-2\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(242))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(242)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)